Subgroups — C₃₇×#19751 #22641004

Order 9472.

684 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#3185	C₂	1	2	yes	maximal
2	C₃₇×#3185	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#3165	C₂	1	2	yes	maximal
4	C₃₇×#3139	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#1723	C₂	1	2	yes	maximal, complemented
6	C₃₇×#1718	C₂	1	2	yes	maximal, complemented
7	C₃₇×#1716	C₂	1	2	yes	maximal, complemented
8	C₃₇×#1707	C₂	1	2	yes	maximal, complemented
9	C₃₇×#1455	C₂	1	1	yes	maximal, characteristic
10	C₃₇×#533	C₂²	1	1	yes	characteristic
11	C₃₇×#533	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
13	C₃₇×#501	—	2	4	no	—
14	C₃₇×#500	C₂²	1	2	yes	—
15	C₃₇×#500	C₂²	1	2	yes	—
16	C₃₇×#498	C₂²	1	2	yes	—
17	C₃₇×#498	C₂²	1	2	yes	—
18	C₃₇×#498	C₂²	1	2	yes	—
19	C₃₇×#498	C₂²	1	1	yes	characteristic
20	C₃₇×#494	C₂²	1	2	yes	—
21	C₃₇×#494	C₂²	1	1	yes	characteristic
22	C₃₇×#491	—	2	2	no	—
23	C₃₇×#487	C₂²	1	2	yes	—
24	C₃₇×#487	C₂²	1	2	yes	—
25	C₃₇×#487	C₂²	1	2	yes	—
26	C₃₇×#487	C₂²	1	2	yes	—
27	C₃₇×#487	C₂²	1	2	yes	—
28	C₃₇×#470	—	2	4	no	—
29	C₃₇×#470	—	2	2	no	—
30	C₃₇×#441	—	2	2	no	—
31	C₃₇×#441	—	2	4	no	—
32	C₃₇×#428	—	2	4	no	—
33	C₃₇×#419	—	2	2	no	—
34	C₄×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
35	C₄×M₁₆×C₃₇	C₂²	1	2	yes	—
36	C₃₇×#322	C₂²	1	2	yes	—
37	C₃₇×#322	C₂²	1	1	yes	characteristic
38	C₈²×C₃₇	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₃₇	—	2	1	no	—
40	C₂×Q₁₆×C₃₇	—	2	2	no	—
41	C₂×Q₁₆×C₃₇	—	2	4	no	—
42	C₂×M₁₆×C₃₇	D₄	1	4	yes	—
43	C₂×M₁₆×C₃₇	D₄	1	2	yes	—
44	C₄⋊Q₈×C₃₇	—	2	2	no	—
45	C₄⋊Q₈×C₃₇	—	2	1	no	—
46	C₄⋊Q₈×C₃₇	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₃₇	C₂³	1	2	yes	—
48	C₄⋊Q₈×C₃₇	C₂³	1	2	yes	—
49	C₄⋊Q₈×C₃₇	C₂³	1	2	yes	—
50	#125×C₃₇	—	2	1	no	—
51	#125×C₃₇	—	2	2	no	—
52	C₂²⋊Q₈×C₃₇	—	2	2	no	—
53	C₂²⋊Q₈×C₃₇	—	2	2	no	—
54	C₂²⋊Q₈×C₃₇	—	2	2	no	—
55	C₄×Q₈×C₃₇	—	2	2	no	—
56	C₄×Q₈×C₃₇	—	2	2	no	—
57	C₄×Q₈×C₃₇	—	2	2	no	—
58	#117×C₃₇	—	2	1	no	—
59	#117×C₃₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
61	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
62	#107×C₃₇	—	2	4	no	—
63	#107×C₃₇	—	2	4	no	—
64	#106×C₃₇	—	2	4	no	—
65	#106×C₃₇	—	2	4	no	—
66	#106×C₃₇	—	2	2	no	—
67	#106×C₃₇	—	2	4	no	—
68	#103×C₃₇	—	2	4	no	—
69	#103×C₃₇	—	2	4	no	—
70	#103×C₃₇	—	2	4	no	—
71	#103×C₃₇	—	2	4	no	—
72	#103×C₃₇	—	2	4	no	—
73	#97×C₃₇	D₄	1	2	yes	—
74	#97×C₃₇	D₄	1	4	yes	—
75	C₄×C₈×C₃₇	C₂³	1	1	yes	characteristic
76	C₄×C₈×C₃₇	C₂³	1	2	yes	—
77	C₄×C₈×C₃₇	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₃₇	C₂³	1	2	yes	—
79	C₂×Q₈×C₃₇	—	2	2	no	—
80	C₂×Q₈×C₃₇	—	2	2	no	—
81	C₂×Q₈×C₃₇	—	2	2	no	—
82	C₂×Q₈×C₃₇	—	2	2	no	—
83	C₂×Q₈×C₃₇	—	4	4	no	—
84	C₂×Q₈×C₃₇	—	4	1	no	—
85	C₂²×C₄×C₃₇	—	4	1	no	—
86	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₃₇	—	4	4	no	—
89	Q₁₆×C₃₇	—	4	8	no	—
90	M₁₆×C₃₇	—	2	8	no	—
91	M₁₆×C₃₇	—	4	2	no	—
92	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
94	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
96	C₄⋊C₄×C₃₇	—	2	2	no	—
97	C₄⋊C₄×C₃₇	—	2	2	no	—
98	C₄⋊C₄×C₃₇	—	2	2	no	—
99	C₄⋊C₄×C₃₇	—	2	2	no	—
100	C₄⋊C₄×C₃₇	—	2	2	no	—
101	C₄⋊C₄×C₃₇	—	2	2	no	—
102	C₄⋊C₄×C₃₇	—	2	2	no	—
103	C₄⋊C₄×C₃₇	—	2	2	no	—
104	C₄⋊C₄×C₃₇	—	4	2	no	—
105	C₄⋊C₄×C₃₇	—	4	2	no	—
106	C₄⋊C₄×C₃₇	—	4	2	no	—
107	C₄⋊C₄×C₃₇	—	4	2	no	—
108	C₂²⋊C₄×C₃₇	—	4	1	no	—
109	C₂²⋊C₄×C₃₇	—	4	2	no	—
110	C₄²×C₃₇	—	4	2	no	—
111	C₄²×C₃₇	—	4	1	no	—
112	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₃₇	C₂×D₄	1	2	yes	—
114	C₄²×C₃₇	C₂⁴	1	1	yes	characteristic
115	C₂³×C₃₇	#127	1	1	yes	characteristic
116	Q₈×C₃₇	—	4	4	no	—
117	Q₈×C₃₇	—	4	4	no	—
118	Q₈×C₃₇	—	4	4	no	—
119	Q₈×C₃₇	—	4	4	no	—
120	Q₈×C₃₇	—	8	2	no	—
121	C₂×C₄×C₃₇	—	2	4	no	—
122	C₂×C₄×C₃₇	—	2	2	no	—
123	C₂×C₄×C₃₇	—	4	2	no	—
124	C₂×C₄×C₃₇	—	4	2	no	—
125	C₂×C₄×C₃₇	—	4	2	no	—
126	C₂×C₄×C₃₇	—	4	2	no	—
127	C₂×C₄×C₃₇	—	8	2	no	—
128	C₂×C₄×C₃₇	#127	1	2	yes	—
129	C₂×C₄×C₃₇	#127	1	1	yes	characteristic
130	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
131	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
132	C₈×C₃₇	—	2	4	no	—
133	C₈×C₃₇	—	2	2	no	—
134	C₈×C₃₇	—	2	2	no	—
135	C₈×C₃₇	#137	1	8	yes	—
136	#19751	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂²×C₃₇	—	2	2	no	—
138	C₂²×C₃₇	—	2	1	no	—
139	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
140	C₄×C₃₇	—	2	2	no	—
141	C₄×C₃₇	—	2	1	no	—
142	C₄×C₃₇	—	8	2	no	—
143	C₄×C₃₇	—	8	2	no	—
144	C₄×C₃₇	—	8	2	no	—
145	C₄×C₃₇	—	8	2	no	—
146	C₄×C₃₇	C₂×#137	1	4	yes	—
147	C₄×C₃₇	#575	1	2	yes	—
148	#3185	C₂×C₃₇	1	2	yes	—
149	#3185	C₂×C₃₇	1	1	yes	characteristic
150	#3165	C₂×C₃₇	1	2	yes	—
151	#3139	C₂×C₃₇	1	1	yes	characteristic
152	#1723	C₂×C₃₇	1	2	yes	complemented
153	#1718	C₂×C₃₇	1	2	yes	complemented
154	#1716	C₂×C₃₇	1	2	yes	complemented
155	#1707	C₂×C₃₇	1	2	yes	complemented
156	#1455	C₂×C₃₇	1	1	yes	characteristic
157	C₂×C₃₇	—	4	1	no	—
158	C₂×C₃₇	#3152	1	2	yes	—
159	C₂×C₃₇	#3153	1	1	yes	characteristic
160	#533	C₂²×C₃₇	1	1	yes	characteristic
161	#533	C₂²×C₃₇	1	2	yes	—
162	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
163	#501	—	2	4	no	—
164	#500	C₂²×C₃₇	1	2	yes	—
165	#500	C₂²×C₃₇	1	2	yes	—
166	#498	C₂²×C₃₇	1	2	yes	—
167	#498	C₂²×C₃₇	1	2	yes	—
168	#498	C₂²×C₃₇	1	2	yes	—
169	#498	C₂²×C₃₇	1	1	yes	characteristic
170	#494	C₂²×C₃₇	1	2	yes	—
171	#494	C₂²×C₃₇	1	1	yes	characteristic
172	#491	—	2	2	no	—
173	#487	C₂²×C₃₇	1	2	yes	—
174	#487	C₂²×C₃₇	1	2	yes	—
175	#487	C₂²×C₃₇	1	2	yes	—
176	#487	C₂²×C₃₇	1	2	yes	—
177	#487	C₂²×C₃₇	1	2	yes	—
178	#470	—	2	4	no	—
179	#470	—	2	2	no	—
180	#441	—	2	2	no	—
181	#441	—	2	4	no	—
182	#428	—	2	4	no	—
183	#419	—	2	2	no	—
184	C₄×M₁₆	C₂²×C₃₇	1	1	yes	characteristic
185	C₄×M₁₆	C₂²×C₃₇	1	2	yes	—
186	#322	C₂²×C₃₇	1	2	yes	—
187	#322	C₂²×C₃₇	1	1	yes	characteristic
188	C₈²	C₂²×C₃₇	1	1	yes	characteristic
189	C₃₇	#19751	1	1	yes	characteristic, direct product factor, Sylow
190	C₂²×Q₈	—	2	1	no	—
191	C₂×Q₁₆	—	2	2	no	—
192	C₂×Q₁₆	—	2	4	no	—
193	C₂×M₁₆	D₄×C₃₇	1	4	yes	—
194	C₂×M₁₆	D₄×C₃₇	1	2	yes	—
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	—	2	1	no	—
197	C₄⋊Q₈	C₂³×C₃₇	1	2	yes	—
198	C₄⋊Q₈	C₂³×C₃₇	1	2	yes	—
199	C₄⋊Q₈	C₂³×C₃₇	1	2	yes	—
200	C₄⋊Q₈	C₂³×C₃₇	1	2	yes	—
201	#125	—	2	1	no	—
202	#125	—	2	2	no	—
203	C₂²⋊Q₈	—	2	2	no	—
204	C₂²⋊Q₈	—	2	2	no	—
205	C₂²⋊Q₈	—	2	2	no	—
206	C₄×Q₈	—	2	2	no	—
207	C₄×Q₈	—	2	2	no	—
208	C₄×Q₈	—	2	2	no	—
209	#117	—	2	1	no	—
210	#117	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
213	#107	—	2	4	no	—
214	#107	—	2	4	no	—
215	#106	—	2	4	no	—
216	#106	—	2	4	no	—
217	#106	—	2	2	no	—
218	#106	—	2	4	no	—
219	#103	—	2	4	no	—
220	#103	—	2	4	no	—
221	#103	—	2	4	no	—
222	#103	—	2	4	no	—
223	#103	—	2	4	no	—
224	#97	D₄×C₃₇	1	2	yes	—
225	#97	D₄×C₃₇	1	4	yes	—
226	C₄×C₈	C₂³×C₃₇	1	1	yes	characteristic
227	C₄×C₈	C₂³×C₃₇	1	2	yes	—
228	C₄×C₈	C₂³×C₃₇	1	1	yes	characteristic
229	C₄×C₈	C₂³×C₃₇	1	2	yes	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	4	4	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
239	Q₁₆	—	4	4	no	—
240	Q₁₆	—	4	8	no	—
241	M₁₆	—	2	8	no	—
242	M₁₆	—	4	2	no	—
243	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
244	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
245	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
246	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	—	4	1	no	—
260	C₂²⋊C₄	—	4	2	no	—
261	C₄²	—	4	2	no	—
262	C₄²	—	4	1	no	—
263	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
264	C₄²	C₂×D₄×C₃₇	1	2	yes	—
265	C₄²	C₂⁴×C₃₇	1	1	yes	characteristic
266	C₂³	#127×C₃₇	1	1	yes	characteristic
267	Q₈	—	4	4	no	—
268	Q₈	—	4	4	no	—
269	Q₈	—	4	4	no	—
270	Q₈	—	4	4	no	—
271	Q₈	—	8	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	8	2	no	—
279	C₂×C₄	#127×C₃₇	1	2	yes	—
280	C₂×C₄	#127×C₃₇	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
282	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
283	C₈	—	2	4	no	—
284	C₈	—	2	2	no	—
285	C₈	—	2	2	no	—
286	C₈	#137×C₃₇	1	8	yes	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	1	no	—
289	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	1	no	—
292	C₄	—	8	2	no	—
293	C₄	—	8	2	no	—
294	C₄	—	8	2	no	—
295	C₄	—	8	2	no	—
296	C₄	C₂×#137×C₃₇	1	4	yes	—
297	C₄	C₃₇×#575	1	2	yes	—
298	C₂	—	4	1	no	—
299	C₂	C₃₇×#3152	1	2	yes	—
300	C₂	C₃₇×#3153	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₇
prime radical	C₂³×C₃₇	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
5. C₃₇×#1723	C₂	semidirect
6. C₃₇×#1718	C₂	semidirect
7. C₃₇×#1716	C₂	semidirect
8. C₃₇×#1707	C₂	semidirect
136. #19751	C₃₇	direct (N × H)
152. #1723	C₂×C₃₇	semidirect
153. #1718	C₂×C₃₇	semidirect
154. #1716	C₂×C₃₇	semidirect
155. #1707	C₂×C₃₇	semidirect
189. C₃₇	#19751	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₇×#3185	2
C₃₇×#3165	2
C₃₇×#3139	2
C₃₇×#1723	2
C₃₇×#1718	2
C₃₇×#1716	2
C₃₇×#1707	2
C₃₇×#1455	2
#19751	37

Subgroups — C37×#19751 #22641004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#19751 #22641004